9 ms·

The Chinese were solving 14th degree polynomials in 1303. Why?

162 points by tombh 8y ago

leereeves 8y ago
At first I thought the headline meant they were factoring or finding the roots of 14th degree polynomials, which would have been quite impressive. But no, the page just says they were calculating the answer for a particular value of x, using "the method of fan fa, today called Horner's method". A simple process, for example: To compute 3x^2 + 12x + 12, start with 3, multiply by x, add 12, multiply the result by x, and add 12 again.
- zrobotics 8y ago
  Agreed, I think the title should be changed to "computing" from solving. Still impressive, but I would think we're not the only people here that headline would confuse.
  - jackhack 8y ago
    "approximating" is the right term. I hate to wordsmith things like this to death, but in speaking about maths, I think it's important to use precise language. To me, "computing" implies an exact calculation. As does "solving". From the article, they did neither, but instead only got practically "close enough." You are right, the headline leads one to conclude something that is not true.
    - betterunix2 8y ago
      "To me, "computing" implies an exact calculation" You are not thinking like a computer scientist: https://en.wikipedia.org/wiki/Computable_number https://en.wikipedia.org/wiki/Computable_number "Computing" typically means finding an approximation up to an arbitrary degree of precision. As for whether or not that counts as "solving," it is a matter of whether or not you will allow lambda expressions in a solution. Think of this: does the quadratic formula count as a solution to quadratic equations? x^2-2=0 has +/-sqrt(2) as a solution, but only if you allow "sqrt" as part of a "solution." If you wanted a decimal representation of the roots you would have to accept an approximation and you would be using some square root finding algorithm. If you are OK with that situation, why not allow a quintic formula that involves lambda expressions? What makes radicals so special? Remember, all the Abel-Ruffini theorem says is that there is no quintic formula involving only arithmetic and radicals -- it leaves open the possibility of another kind of formula for the quintic.
    - ummonk 8y ago
      Computing is an algorithm which can produce the value to any given precision (I.e. bounds size). Approximating is an algorithm which eventually converges to the true value but you never know when you’ve actually gotten close enough. For this reason the computable numbers are a subset of the approximately numbers.
- abhgh 8y ago
  Same here. Given Abel's Impossibility theorem [1], the headline got me very curious. Not that you couldn't solve for any case (the theorem talks about a general solution), but still interesting. [1] http://mathworld.wolfram.com/AbelsImpossibilityTheorem.html http://mathworld.wolfram.com/AbelsImpossibilityTheorem.html
  - fspeech 8y ago
    Solving particular equations are definitely possible. Abel's theorem says there is no general formula to solve the quintic and higher order polynomials.
    - abhgh 8y ago
      Yeah, I realized that this would be potentially be confusing about what I meant, so I edited my comment.
  - octachron 8y ago
    Abel-Ruffini theorem only speaks about general solution in term of elementary operations and nth-root, which is a very restricted form of solution. Numerically solving roots for generic polynomial is perfectly doable (and the nth-root already require approximate computation). Moreover, the radical solution for cubic and quartic equation is already quite difficult and inefficient to use compared to numerical solution.
- fspeech 8y ago
  No the headline was not wrong. One of the answers explained Horner's method and its relation to using abacus (which has a small register file, in modern lingo). However that is not the extent of the book mentioned. You can find the book here: https://zh.wikipedia.org/wiki/%E5%9B%9B%E5%85%83%E7%8E%89%E9%89%B4 https://zh.wikipedia.org/wiki/%E5%9B%9B%E5%85%83%E7%8E%89%E9... The equations dealt with have been transcribed into modern algebraic expressions. They were definitely solved, not merely computed. The more complete answer should have been "The higher order polynomial (as high as 14) came from solving multivariate simultaneous equations (with degrees as high as 4) by elimination of variables."
  - wbhart 8y ago
    I'm not seeing that. I see a degree 10 equation, which oddly has no rational roots (though my guess is this is what is meant by "solving"). I've never seen any mention of the Chinese solving degree 14 equations in any books on the history of mathematics. I'm very dubious of the claim, in any form, other than finding rational roots of polynomials (which the Greeks were doing more than a millennium earlier). Edit: no they were finding rational approximations to roots, which is actually pretty clever.
    - fspeech 8y ago
      "Edit: no they were finding rational approximations to roots, which is actually pretty clever." Right they provided a general numerical method that can solve univariate polynomials of any degree. See https://zh.wikipedia.org/wiki/%E7%A7%A6%E4%B9%9D%E9%9F%B6%E7%AE%97%E6%B3%95 https://zh.wikipedia.org/wiki/%E7%A7%A6%E4%B9%9D%E9%9F%B6%E7... "南宋数学家秦九韶将贾宪的增乘开方术推广，以求解任意高次方程的实数根的数值解"
  - leereeves 8y ago
    I wasn't referring to the reddit page, but the Wikipedia page it's based on: > Si-yüan yü-jian (《四元玉鑒》), or Jade Mirror of the Four Unknowns, was written by Zhu Shijie in 1303 AD and marks the peak in the development of Chinese algebra. The four elements, called heaven, earth, man and matter, represented the four unknown quantities in his algebraic equations. It deals with simultaneous equations and with equations of degrees as high as fourteen. The author uses the method of fan fa, today called Horner's method, to solve these equations.[48] If there's more to the story you might want to correct Wikipedia.
maroonmed 8y ago
Because Asians are good at math, duh.
betterunix2 8y ago
One of the things I find interesting about ancient Chinese math is how computational it was (in modern terminology we would say they were only interested in computable numbers); AFAIUI the ancient Chinese were vaguely aware of numbers that could not be computed exactly (i.e. irrationals) but mostly cared about being able to compute approximations up to whatever precision was required for a given task. So while Western mathematicians were struggling in vain to find a way to double a cube using a compass and straightedge or to derive a quintic formula, the Chinese mathematicians were satisfied with an algorithm that found useful approximations. There is a profound difference in philosophy, as evidenced by the title of this entire thread: the Chinese were "solving" 14th degree polynomials, but only in the sense of having an algorithm for doing so (as opposed to having a formula like the quadratic formula; in fact there is no such formula for degree 14 polynomials, or for any polynomials beyond the quartic).
- how_to_bake 8y ago
  Hate to be annoying, but I'm a native English speaker and I just skipped over "AFAIUI" because it was too much work to guess what it meant. "As far as I understand it" ? RIP if you're not a native speaker.
  - twtw 8y ago
    RIP if you are a native speaker too.
    - Asooka 8y ago
      IRC-speak is its own language :)
      - deleted 8y ago
        [deleted]
      - plussed_reader 8y ago
        A predecessor of pager code.
samfriedman 8y ago
AskHistorians is surely one of the great jewels of Reddit. Fantastically moderated, and filled with an array of helpful experts happy to research and write up answers to all sorts of interesting questions and topics.

162 points by tombh 8y ago

At first I thought the headline meant they were factoring or finding the roots of 14th degree polynomials, which would have been quite impressive. But no, the page just says they were calculating the answer for a particular value of x, using "the method of fan fa, today called Horner's method". A simple process, for example: To compute 3x^2 + 12x + 12, start with 3, multiply by x, add 12, multiply the result by x, and add 12 again.

zrobotics 8y ago
Agreed, I think the title should be changed to "computing" from solving. Still impressive, but I would think we're not the only people here that headline would confuse.
- jackhack 8y ago
  "approximating" is the right term. I hate to wordsmith things like this to death, but in speaking about maths, I think it's important to use precise language. To me, "computing" implies an exact calculation. As does "solving". From the article, they did neither, but instead only got practically "close enough." You are right, the headline leads one to conclude something that is not true.
  - betterunix2 8y ago
    "To me, "computing" implies an exact calculation" You are not thinking like a computer scientist: https://en.wikipedia.org/wiki/Computable_number https://en.wikipedia.org/wiki/Computable_number "Computing" typically means finding an approximation up to an arbitrary degree of precision. As for whether or not that counts as "solving," it is a matter of whether or not you will allow lambda expressions in a solution. Think of this: does the quadratic formula count as a solution to quadratic equations? x^2-2=0 has +/-sqrt(2) as a solution, but only if you allow "sqrt" as part of a "solution." If you wanted a decimal representation of the roots you would have to accept an approximation and you would be using some square root finding algorithm. If you are OK with that situation, why not allow a quintic formula that involves lambda expressions? What makes radicals so special? Remember, all the Abel-Ruffini theorem says is that there is no quintic formula involving only arithmetic and radicals -- it leaves open the possibility of another kind of formula for the quintic.
  - ummonk 8y ago
    Computing is an algorithm which can produce the value to any given precision (I.e. bounds size). Approximating is an algorithm which eventually converges to the true value but you never know when you’ve actually gotten close enough. For this reason the computable numbers are a subset of the approximately numbers.
abhgh 8y ago
Same here. Given Abel's Impossibility theorem [1], the headline got me very curious. Not that you couldn't solve for any case (the theorem talks about a general solution), but still interesting. [1] http://mathworld.wolfram.com/AbelsImpossibilityTheorem.html http://mathworld.wolfram.com/AbelsImpossibilityTheorem.html
- fspeech 8y ago
  Solving particular equations are definitely possible. Abel's theorem says there is no general formula to solve the quintic and higher order polynomials.
  - abhgh 8y ago
    Yeah, I realized that this would be potentially be confusing about what I meant, so I edited my comment.
- octachron 8y ago
  Abel-Ruffini theorem only speaks about general solution in term of elementary operations and nth-root, which is a very restricted form of solution. Numerically solving roots for generic polynomial is perfectly doable (and the nth-root already require approximate computation). Moreover, the radical solution for cubic and quartic equation is already quite difficult and inefficient to use compared to numerical solution.
fspeech 8y ago
No the headline was not wrong. One of the answers explained Horner's method and its relation to using abacus (which has a small register file, in modern lingo). However that is not the extent of the book mentioned. You can find the book here: https://zh.wikipedia.org/wiki/%E5%9B%9B%E5%85%83%E7%8E%89%E9%89%B4 https://zh.wikipedia.org/wiki/%E5%9B%9B%E5%85%83%E7%8E%89%E9... The equations dealt with have been transcribed into modern algebraic expressions. They were definitely solved, not merely computed. The more complete answer should have been "The higher order polynomial (as high as 14) came from solving multivariate simultaneous equations (with degrees as high as 4) by elimination of variables."
- wbhart 8y ago
  I'm not seeing that. I see a degree 10 equation, which oddly has no rational roots (though my guess is this is what is meant by "solving"). I've never seen any mention of the Chinese solving degree 14 equations in any books on the history of mathematics. I'm very dubious of the claim, in any form, other than finding rational roots of polynomials (which the Greeks were doing more than a millennium earlier). Edit: no they were finding rational approximations to roots, which is actually pretty clever.
  - fspeech 8y ago
    "Edit: no they were finding rational approximations to roots, which is actually pretty clever." Right they provided a general numerical method that can solve univariate polynomials of any degree. See https://zh.wikipedia.org/wiki/%E7%A7%A6%E4%B9%9D%E9%9F%B6%E7%AE%97%E6%B3%95 https://zh.wikipedia.org/wiki/%E7%A7%A6%E4%B9%9D%E9%9F%B6%E7... "南宋数学家秦九韶将贾宪的增乘开方术推广，以求解任意高次方程的实数根的数值解"
- leereeves 8y ago
  I wasn't referring to the reddit page, but the Wikipedia page it's based on: > Si-yüan yü-jian (《四元玉鑒》), or Jade Mirror of the Four Unknowns, was written by Zhu Shijie in 1303 AD and marks the peak in the development of Chinese algebra. The four elements, called heaven, earth, man and matter, represented the four unknown quantities in his algebraic equations. It deals with simultaneous equations and with equations of degrees as high as fourteen. The author uses the method of fan fa, today called Horner's method, to solve these equations.[48] If there's more to the story you might want to correct Wikipedia.

maroonmed 8y ago

Because Asians are good at math, duh.

betterunix2 8y ago

One of the things I find interesting about ancient Chinese math is how computational it was (in modern terminology we would say they were only interested in computable numbers); AFAIUI the ancient Chinese were vaguely aware of numbers that could not be computed exactly (i.e. irrationals) but mostly cared about being able to compute approximations up to whatever precision was required for a given task. So while Western mathematicians were struggling in vain to find a way to double a cube using a compass and straightedge or to derive a quintic formula, the Chinese mathematicians were satisfied with an algorithm that found useful approximations. There is a profound difference in philosophy, as evidenced by the title of this entire thread: the Chinese were "solving" 14th degree polynomials, but only in the sense of having an algorithm for doing so (as opposed to having a formula like the quadratic formula; in fact there is no such formula for degree 14 polynomials, or for any polynomials beyond the quartic).

how_to_bake 8y ago
Hate to be annoying, but I'm a native English speaker and I just skipped over "AFAIUI" because it was too much work to guess what it meant. "As far as I understand it" ? RIP if you're not a native speaker.
- twtw 8y ago
  RIP if you are a native speaker too.
  - Asooka 8y ago
    IRC-speak is its own language :)
    - deleted 8y ago
      [deleted]
    - plussed_reader 8y ago
      A predecessor of pager code.

samfriedman 8y ago

AskHistorians is surely one of the great jewels of Reddit. Fantastically moderated, and filled with an array of helpful experts happy to research and write up answers to all sorts of interesting questions and topics.